Papers

Let $f$ be a positive integer, and let $\chi$ be a primitive quadratic character of conductor $f$. Let $k$ be a positive integer, and write $B_k(\chi,X)$ for the $k$-th Bernoulli polynomial corresponding to $\chi$. Suppose $B_k(\chi,X)$ is irreducible and of degree at least $2$. Then for 100% of positive integers $m$ divisible by $f$, the Diophantine equation \[ \chi(1) \cdot (x+1)^k+\chi(2) \cdot (x+2)^k+\cdots+\chi(m) \cdot (x+m)^k \, =\, y^n, \] has no solutions with $x$, $y$, $n$ integers, and $n \ge 2$.

We study time-discrete parametric approximations of evolution equations in Hilbert spaces based on residual minimization. The solution is represented by a parametrized ansatz belonging to a low-dimensional nonlinear manifold, and time stepping is performed by minimizing suitably defined residuals at each step. Two natural residual formulations are considered: discretization followed by parametrization of the evolution equation, and discretization of the Dirac–Frenkel variational principle governing the parameter dynamics. A unified error analysis is developed for both approaches within the family of $\zeta$-methods. The resulting bounds separate the effects of time discretization from those of residual minimization and yield first- and second-order convergence under Lipschitz, one-sided Lipschitz, and dissipativity assumptions. For the variational formulation, additional stability conditions involving the conditioning of the parametrization map arise naturally. The framework is applied to Gaussian approximation manifolds, for which residual norms and gradients admit explicit closed-form expressions when polynomial operators are involved. This enables efficient implementation without spatial discretization. Numerical experiments for time-dependent Schrödinger equations illustrate the theoretical convergence rates and the influence of residual accuracy on conservation properties.

Let $\mathscr{A}(X)$ and $\mathscr{K}(X)$ denote the ideals of approximable and compact operators on a Banach space $X$, respectively. We construct a unital Banach algebra $A$ of density character $\mathfrak c=2^{\aleph_0}$, with exactly one non-zero proper closed two-sided ideal, such that, for every Banach space $X$, the algebra $A$ is isomorphic to neither $\mathscr{B}(X)/\mathscr{A}(X)$ nor $\mathscr{B}(X)/\mathscr{K}(X)$. Thus $A$ is not a Calkin algebra under either of the two customary conventions.

Given a compact CY3 $Y$, its A-side (resp. B-side) universal intermediate Jacobian $X$ (resp. $X^{!}$) admits a natural hyperkahler structure. Both $X$ and $X^{!}$ determines 3d Rozansky-Witten theories, in both A-model and B-model. We describe some surprising 3d mirror symmetry phenomena between $X$ and $X^{!}$. This includes (i) a 3d SYZ construction of $X^{!}$ from $X$ via moduli of certain 3d A-branes in $X$ constructed from $D^{b}\left( Y,\Omega \right) $ with varying stability conditions given by $\varpi $; (ii) The 3d B-brane on $X^{!}$, constructed from varying $D^{b}\left( Y,\Omega \right) $ with fixed $\varpi $, is realized as the 3d SYZ transformation of a cotangent fiber 3d A-brane in $X$. Under mirror symmetry between $Y$ and $Y^{\vee }$, the roles for $X$ and $% X^{!}$ got interchanged. We also construct 3d A- and B-branes in $X$ and $X^{!}$ via DT theory and discrete symmetries on $Y$ and $Y^{\vee }$.

A graph $G$ with $m$ edges is said to be a Nosal graph if $\rho(G)>\sqrt{m}$. For a graph $G$, we write $bk(G)$ for its maximum book size and $\tau(G)$ for the number of edges contained in triangles. Li, Liu and Zhang [J. Combin. Theory Ser. B 179 (2026) 219–249] proved that every $m$-edge Nosal graph satisfies $bk(G)> \frac{1}{24}\sqrt{m}$ and $\tau(G) > \frac{1}{12}\sqrt{m}$. Recently, two results on the booksize constant are proved: $\frac{1}{9}$ by Zhai, Li and Lou [arXiv:2601.10163v2], and $\frac{1}{4}$ by Chen, Li and Tang [arXiv:2607.16746v1]. In this paper, we establish the following result: Every $m$-edge graph $G$ with no isolated vertices and $\rho(G)\geq \sqrt{m}$ that is not isomorphic to any complete bipartite graph satisfies $bk(G)\geq\frac{\rho(G)}{3}$ and $\tau(G)\geq \rho(G)$. As direct consequences, we answer a question of Li, Liu and Zhang [J. Combin. Theory Ser. B 179 (2026) 219–249] and confirm a conjecture of Li, Feng and Peng [J. Graph Theory 110 (4) (2025) 408–425].

We investigate the Casimir effect for a massive real scalar field confined between two perfectly reflecting parallel plates in the presence of a position-dependent effective mass, a mechanism for coupling a scalar background to a scalar field. Exact normal modes are obtained by solving the corresponding Klein-Gordon equation, leading to a transverse energy spectrum that exhibits a characteristic Landau-like structure despite the absence of an external magnetic field. Upon quantization of the field, the vacuum energy is evaluated by means of generalized zeta-function regularization together with an appropriate renormalization procedure. The renormalized vacuum energy naturally separates into a Landau-like contribution and an additional term induced by the spatial dependence of the effective mass. We show analytically and numerically that both contributions are exponentially suppressed in the strong-coupling regime. In the opposite limit, the Landau-like contribution smoothly reproduces the standard vacuum energy for a massive scalar field confined between parallel plates, whereas the additional contribution becomes singular owing to the restricted domain of validity of the exact spectrum. Except in the vicinity of this singular limit, the vacuum energy is shown to be dominated by the Landau-like sector. Our results establish a direct connection between position-dependent effective masses and boundary-induced quantum vacuum phenomena, providing a new exactly solvable framework for the investigation of Casimir effects in spatially inhomogeneous relativistic systems.

Given a solution of the Brent equations, a partial solution can be substituted into the Brent equations which makes it a reduced polynomial system. Then the reduced polynomial system can be solved by symbolic software, and a parameterized solution set can often be found. However, the solution set admits a positive-dimensional isotropy group action. The parameterized solution set may lie entirely in a single isotropy group orbit. In this paper, we give a geometric explanation of making the substitution. Considering the foliation structure of the solution set, a method on choosing the partial solution is proposed. By the proposed method, parameterized solution sets that intersect distinct isotropy group orbits can be efficiently determined. In particular, the rational solution found by Dumas, Pernet and Sedoglavic is parameterized. In the parameterized solution set, we can find infinitely many inequivalent 48-multiplication algorithms with only rational coefficients.

In this paper, we propose an adaptive physics-preserving multiscale method for incompressible and immiscible two-phase flow in high-contrast porous media. The method couples a physics-preserving implicit-pressure explicit-saturation scheme (P-IMPES) with the mixed constraint energy minimizing generalized multiscale finite element method. The core algorithmic component is an adaptive update strategy for the saturation-dependent coefficient. Since the effective permeability \(\kappa_n=\lambda_t(S_w^n)K\) depends on the evolving saturation through the total mobility, we introduce an adaptive update algorithm that monitors the variation of the mobility-weighted coefficient and regenerates the multiscale spaces only when a prescribed tolerance is exceeded. A local postprocessing step is further used to recover fine-grid mass conservation. The analysis is a central part of the paper. We prove local conservation for both phases, the unbiased property of the phase formulation, and bounds preservation under a suitable CFL condition. For the advection-dominated case, we establish velocity and saturation error estimates, which clearly identify the contributions from the adaptive tolerance, the coarse mesh size, the spectral approximation, and the front-layer error. Numerical experiments on different high-contrast permeability fields confirm the physical properties of the method and show that smaller adaptive tolerances improve the saturation approximation while avoiding unnecessary updates of the multiscale spaces.

We produce new collections of homotopy 4-spheres and homotopy real projective 4-spaces by extending a classical construction due to Cappell-Shaneson. We use the extension to devise a mechanism that produces sets of smooth 4-manifolds within a given homeomorphism class that satisfies a mild topological condition.

Foster and Viswanathan (1996) extend the discrete-time setting of Kyle (1985) to multiple informed traders who have partial information about the stock's terminal dividend. We resolve two long-standing open problems in this literature. First, we prove that an equilibrium exists in the setting of Foster and Viswanathan (1996). Second, as the number of trading times goes to infinity, we prove that the discrete-time equilibrium converges to the continuous-time equilibrium already proven to exist in Back, Cao, and Willard (2000).

This paper addresses the robust control of switched systems under arbitrary switching with performance guarantees. We propose a framework that jointly synthesizes a feedback policy and a certified upper bound on its corresponding infinite-horizon closed-loop value function. The proposed upper bound not only certifies the performance of the synthesized policy, but can also be optimized during controller synthesis. More precisely, our approach associates functions with the nodes of a path-complete graph and enforces graph-based Bellman inequalities along its edges. Exploiting a newly introduced notion of reachability graph, these functions are combined into both a feedback policy and a certified upper bound on its corresponding closed-loop value function, expressed as a pointwise min-max combination of the graph-indexed functions. For linear switched systems with quadratic stage costs, the proposed framework admits tractable computational formulations based on semidefinite programming and alternating optimization. Numerical experiments, including a building temperature regulation benchmark, demonstrate the practical usefulness of the proposed approach both for direct feedback control using the synthesized policy and for model predictive control using the certified upper bound as a terminal cost.

We study the $L^p$ Neumann problem for the Stokes system on bounded Lipschitz domains in $\mathbb R^n$ with $n\ge 2$. Our main contribution is to introduce a nonlinear gradient quantity – namely, the linear gradient weighted by a suitable power of the pair $(\nabla u,\phi)$ – in place of the standard linear gradient used in previous work by Geng and Shen (2025). This new approach allows us to establish a global second-order estimate, which in turn yields an improved reverse Hölder inequality and extends the known range of solvability for convex domains, particularly improving the upper bound for $n\ge 3$. Beyond the convex setting, our method also applies to semi-convex domains. Moreover, for more general Lipschitz domains, we prove solvability under a smallness condition on the second fundamental form of the boundary, assuming the boundary has second-order derivatives in the weak-type Lorentz spaces $W^2L^{n-1,\infty}$ for $n\ge 3$, or $W^2L^{1,\infty}\log L$ for $n=2$. In particular, our results cover all $W^{2,q}$ domains with $q>n-1$.

In this paper, we construct double $L$-values of negative weight that admit series representations. In our construction, we use the Thue-Morse sequence, which appears in combinatorics on words and automatic sequences.

Principal component analysis (PCA) is optimal for the linear reconstruction of Gaussian data, a foundational property underlying its central role in algorithms and signal processing. Its nonlinear analogue, however, is notoriously subtle: in 2011, Mallat and Zeitouni conjectured that the Karhunen–Loève (KL) basis remains optimal even when the retained coordinates are chosen adaptively per sample, a property that would theoretically justify the ubiquitous pipeline of PCA followed by sparse thresholding. In this paper, we establish a $1+O(1/\sqrt{d})$-approximate version of the retained-energy form of the Mallat–Zeitouni conjecture, showing that the KL basis is within this factor of the optimal basis. This dimension-free comparison depends only on the number of retained coordinates and shows that the possible advantage of optimizing over all orthonormal bases vanishes as $d$ grows. It complements the universal-constant reconstruction-error comparison of Litvak and Tikhomirov (Ann. Appl. Probab., 2018), while providing a comparison naturally suited for algorithmic analysis. Our proof rests on a clean, conceptual reduction: we relax arbitrary rotations to a deterministic threshold bound via Schur–Horn majorization, and identify the remaining loss with the correlation gap of the rank-$d$ uniform matroid over Gaussian level sets.

We present an extension of the template-based approach for Schur numbers developed by Rowley. This new form of template construction, which we call shifted S-templates, was discovered during a conversation with ChatGPT 5.5 Pro, then refined, verified, and extended to multiple shifted S-templates. These new templates generalize the first ones by giving more flexibility in the coloring. Using this added flexibility with the new way to color the special label cells of the template, we exhibit a template which yields the recurrence $S(k+2) \geq 10S(k)+2$, improving on the classical Abbott-Hanson recurrence $S(k+2) \geq 9S(k)+4$ for the same step. Combined with the known bounds $S(6) \geq 536$ and $S(11) \geq 203\,828$, this implies $S(8) \geq 5\,362$ and $S(13) \geq 2\,038\,282$, improving the previously listed lower bounds $5\,286$ and $2\,011\,290$.

Model checking is the automated verification of properties (specified in some modal logic) in labeled transition systems (LTSs); it is an essential tool in ensuring software systems function as intended. State spaces of software grow exponentially, and heuristics are needed to ensure model checking remains feasible in real-world applications. Heuristics, in turn, require a good understanding on the typical behaviour of LTSs. In this paper, we use random graph theory to create a probabilistic model of large LTSs. From a theoretical analysis of the creation of large LTSs, backed by empirical data from the Model Checking Contest, we endow these models with realistic parameter values. Then, we analyze the asymptotic behaviour of this model under LTL and CTL, two modal logics popular in model checking. We show that, depending on the precise model, as the size grows to infinity we either have a convergence law (for every formula, the probability that it holds converges to a limit) or a 0-1 law (...and this limit is 0 or 1). We also discuss the theoretical complexity of determining these limits, and give algorithms for doing so. These results are the starting point towards a deep theoretical understanding of typical LTS behaviour, and highlight the promising applicability of random graph theory to model checking. \keywords{Model checking \and Random graphs \and 0-1 laws

Roman domination and its variants form an important family of domination-type graph parameters motivated by protection, fault tolerance, and resource allocation. A Roman dominating function of a graph \(G\) is a function \(f:V(G)\rightarrow\{0,1,2\}\) such that every vertex \(v\) with \(f(v)=0\) has a neighbour \(u\) with \(f(u)=2\). The weight of \(f\) is \(w(f)=\sum_{v\in V(G)}f(v)\), and the minimum weight of a Roman dominating function of \(G\) is the Roman domination number, denoted by \(\gamma_R(G)\). In this paper, we study four variants of Roman domination on two natural subclasses of bipartite graphs, namely convex bipartite graphs and chordal bipartite graphs. On the positive side, we develop a unified left-to-right dynamic programming framework for Roman-\(\{2\}\) domination, double Roman domination, perfect Roman domination, and unique response Roman domination on convex bipartite graphs. The algorithms exploit the interval structure of one bipartition class and represent all unfinished requirements using a constant number of boundary indices. Consequently, each of the four parameters can be computed in \(O(n^6)\) time, where \(n=|V(G)|\). On the negative side, we prove that Roman-\(\{2\}\) domination, perfect Roman domination, and unique response Roman domination remain NP-complete on chordal bipartite graphs. These results establish a clear algorithmic separation between convex bipartite graphs, where the interval ordering yields polynomial-time solvability, and the broader class of chordal bipartite graphs, where several Roman-type domination problems remain computationally intractable.

The least-squares splitting algorithm for the Monge-Ampère equation has been used successfully in computations for several years, but a convergence theory for fully discrete splitting schemes of this type has remained unavailable. In this work, we introduce and analyze a finite element framework for smooth solutions of the Dirichlet Monge-Ampère equation in two dimensions. The proposed schemes combine a discrete Hessian reconstruction with a local projection onto the determinant constraint. Under a discrete Miranda-Talenti estimate and standard approximation properties of the Hessian reconstruction, we prove local convergence of the iterative scheme and optimal-order convergence of its limit to the exact solution in an $H^2$-type norm. We verify the estimates for conforming $C^1$ schemes, including the Argyris element, and for $C^0$-interior penalty and DG schemes of degree at least three; quadratic $C^0$-interior penalty and DG schemes are also covered when $|u|_{H^3(\Omega)}$ is sufficiently small. To the best of our knowledge, these discretizations have not previously been proposed or analyzed for least-squares splitting methods. Numerical experiments confirm the theoretical convergence rates.

This study applied a mathematical tool from Topological Data Analysis (TDA), called the Mapper algorithm, to gene expression data from more than 1,000 TCGA-BRCA patients to identify hidden molecular patterns associated with survival. Patients located near high-risk regions of the network showed significantly poorer survival, and highly proliferative gene expression patterns were associated with worse outcomes overall, although treatment narrowed this survival gap across proliferation groups. The analysis further uncovered patients whose survival outcomes were inconsistent with their expected clinical behavior, including a subgroup of Basal-like patients with unexpectedly favorable outcomes linked to a distinct, more treatment-responsive gene signature, revealing molecular programs missed by traditional classification methods. Validation through training and testing on unseen patients confirmed that topology-derived risk groups remained significantly associated with survival after adjusting for age, tumor stage, and treatment, demonstrating that the geometric structure of gene expression data contains clinically meaningful prognostic information beyond traditional breast cancer classification methods.

We study Heilbronn's triangle problem in the unit right triangle, where $n$ points are placed to maximize the smallest of the $\binom{n}{3}$ triangle areas they span. We prove a boundary-structure result: unless all three vertices are occupied, some optimal configuration with $n \ge 5$ has at least four points on the boundary, one edge containing two of them. With the affine $S_3$ symmetry this fixes four boundary points and $n$ orientation variables in a mixed-integer model that certifies global optimality for all $n \le 8$ up to a relative gap of $10^{-4}$. For $n \le 7$ this independently confirms the known exact optima, the case $n = 7$ having been proved by Zeng and Chen with symbolic computation. For $n = 8$, where only bounds were known, it narrows the gap to Cantrell's configuration fourfold in 39 minutes on one machine. The optimum there is conjectured to be the real root of a septic obtained by Chen, Zeng and Zhou, which our reconstruction confirms to 250 digits. We show its Galois group is $S_7$, so on that conjecture no expression in radicals exists.

In this paper, we introduce a mathematical structure called the quasiPfaffian. The quasiPfaffian is analogous to the quasideterminant, a structure used instead of a determinant in noncommutative settings. Building on the Sylvester identity for the quasiPfaffian, we develop a novel transformation, named the Sylvester Moutard transform, this generates new solutions for Moutard transformable integrable systems, such as the Novikov Veselov equation and the two dimensional sine Gordon equation. We also briefly review the classical Moutard transformation in the context of quasiPfaffians and discuss several additional properties of this new object.

A positive integer $m$ is termed a Hall number if every finite group $G$ whose order is precisely divisible by $m$ possesses a Hall subgroup of order $m$. Seeking generalizations of Sylow's theorem and Hall's theorem for finite solvable groups, Jiping Zhang asked for a full classification of Hall numbers, a problem recently solved by Guo, Hu and Li. Inspired by Zhang's problem, Guohua Qian put forward an analogous problem on a full classification of Hall complement numbers. Recall that a positive integer $m$ is called a Hall complement number provided that every finite group $G$ with $m$ precisely dividing $|G|$ admits a Hall subgroup of order $|G|/m$. In the present paper, we prove that every Hall complement number is either $1$ or of the form $4k+2$ for some non-negative integer $k$, thus answering Qian's problem.

We show sharpness of the phase transition for a nearest-neighbour percolation model on $\mathbb Z^d$, where vertices carry independent types and the percolation probability of edges depends on the type of the adjacent vertices. Our proof uses the OSSS inequality and adapts to our setup the method developed in Duminil-Copin et al. (2017) for the random cluster model. Additionally, we provide a more extensive study of the special case of combined Bernoulli bond and site percolation featuring a phase transition with two parameters.

We study local integrability of gradients of singular interaction kernels in aggregation equations. Suppose near the singularity that $|\nabla K|\asymp|\nabla f|\,|f|^{-(\kappa+1)}$, where $\kappa>0$ and $f$ is real-analytic with $f(0)=0$. A real log-resolution of $f$ and its Jacobian ideal gives the exact criterion $\nabla K\in L^p_{\mathrm{loc}}$ if and only if $p<p^*(\kappa)$, where \[ p^*(\kappa)=\min_i\frac{a_i+1}{(\kappa+1)\nu_i-M_i}. \] Here $\nu_i$, $M_i$, and $a_i$ are the vanishing orders of $f$, those of its Jacobian ideal, and the discrepancy exponents. Moreover, $\operatorname{rlct}_0(f)/(\kappa+1)\leq p^*(\kappa)$, with equality exactly when an RLCT-computing divisor has $M_i=0$; in particular, the inequality is strict for isolated zeros in dimension at least two. For quasi-homogeneous singularities with isolated real zero, $p^*(\kappa)=|w|/(\kappa d+\max_i w_i)$. Finally, if $p^*(\kappa)>1$, a suitable global truncation gives $\nabla K\in L^1(\mathbb R^n)$, yielding the kernel hypothesis in the Morrey-space well-posedness theorem of \cite{suleiman2023existence}.

Stabilizer codes are often constructed within the Calderbank–Shor–Steane (CSS) framework, where two mutually orthogonal binary classical codes define $X$ and $Z$-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with $X$-, $Y$-, and $Z$-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the $Y$-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ$^2$ hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.

For the Selberg zeta function of geometrically finite infinite-area hyperbolic orbisurfaces with twists by finite-dimensional unitary representations, we establish a factorization formula in terms of a Weierstrass product of the Laplace resonances of the considered hyperbolic orbisurface, Barnes G-functions, gamma functions, and the singularity degrees of the representation. We thereby provide an interpretation of the zeros and poles of the Selberg zeta function by spectral and geometric entities of the orbisurface and the representation. This formula generalizes the factorization result by Borthwick, Judge and Perry to hyperbolic orbisurfaces with orbifold singularities as well as to unitary twists. Also in the untwisted case, the presence of orbifold singularities yields a separate, previously unobserved contribution to the factorization formula.

We study the boundary asymptotic behavior of Bergman functions on planar domains. Motivated by Chen's question on the equivalence between uniform perfectness of the boundary and the sharp growth rates of the Bergman kernel and the Bergman metric, we focus on the second part of the question concerning the Bergman metric. We prove that $\partial\Omega$ is uniformly perfect if and only if $K_{\Omega}^{(1)}(w)\asymp \delta_{\Omega}(w)^{-4}$. We also find that under suitable weak uniform perfectness conditions, there exist sequences of points along which $b_{\Omega}(w_n)=o(\delta_{\Omega}(w_n)^{-1})$, providing partial evidence toward an affirmative answer. Our method relies on sharp lower and upper bounds for $K_{\Omega}^{(1)}$ and $K_{\Omega}$. As an application, we obtain corresponding lower bounds for the Bergman distance on certain planar domains.

Motivated by batch and semi-batch process operation, we study finite-horizon open-loop dynamic optimization problems with uncertain parameters. A common computational approach replaces the expected performance criterion by an average over finitely many sampled parameter realizations. We develop a statistical theory for the resulting sample-based optimal value as an estimator of the population optimal value. The analysis is based on a stability estimate showing that terminal losses depend Lipschitz continuously on the time-integrated control, which records the cumulative input delivered up to each time. This estimate yields a functional central limit theorem for the sample-based objective and a statistical limit theorem for the corresponding optimal value error. As a consequence, we obtain confidence intervals for the population optimal value. When the population optimizer is unique, the limit is Gaussian and leads to a plug-in confidence interval. When multiple optimal policies may exist, we use a subsampling confidence interval that does not require uniqueness. The methodology is illustrated on two fed-batch case studies in which feed-rate profiles are optimized under parametric uncertainty.

An uncertainty principle due to H.S. Shapiro, the so-called Umbrella Theorem, asserts that there is no square integrable function uniformly dominating all the elements of an orthonormal family in $L^2(\mathbb{R})$ and their Fourier transforms, unless the sequence is finite. In this paper we present some results on Umbrella Theorems in $L^2(\mathbb{R}^d)$ related to time-frequency representations. We further extend the analysis to the case of $L^2(\mathbb{R}^+)$, by means of the Mellin transform.

A sequence $(a_n)_{n \in \mathbb{N}}$ of non-negative real numbers is called log-concave at $n$ if $a_n^2 \geq a_{n+1}a_{n-1}$. This property has been generalised in various ways to families of polynomials. We introduce a new variant and show that certain types of D'Arcais polynomials have the respective properties at certain points.